You have found the following ages (in years) of 6 sloths. Those sloths were randomly selected from the 39 sloths at your local zoo: $ 17,\enspace 10,\enspace 1,\enspace 17,\enspace 11,\enspace 14$ Based on your sample, what is the average age of the sloths? What is the standard deviation? You may round your answers to the nearest tenth.
Solution: Because we only have data for a small sample of the 39 sloths, we are only able to estimate the population mean and standard deviation by finding the sample mean $({\overline{x}})$ and sample standard deviation $({s})$ To find the sample mean , add up the values of all $6$ samples and divide by $6$ $ {\overline{x}} = \dfrac{\sum\limits_{i=1}^{{n}} x_i}{{n}} = \dfrac{\sum\limits_{i=1}^{{6}} x_i}{{6}} $ $ {\overline{x}} = \dfrac{17 + 10 + 1 + 17 + 11 + 14}{{6}} = {11.7\text{ years old}} $ Find the squared deviations from the mean for each sample. Since we don't know the population mean, estimate the mean by using the sample mean we just calculated {28.09} + {2.89} + {114.49} + {28.09} + {0.49} + {5.29}} {{6 - 1}} $ {s^2} = \dfrac{{179.34}}{{5}} = {35.87\text{ years}^2} $ As you might guess from the notation, the sample standard deviation $({s})$ is found by taking the square root of the sample variance $({s^2})$ ${s} = \sqrt{{s^2}}$ $ {s} = \sqrt{{35.87\text{ years}^2}} = {6\text{ years}} $ We can estimate that the average sloth at the zoo is 11.7 years old. There is also a standard deviation of 6 years.